A computational method for multiple steady Hele-Shaw bubbles in planar domains

TL;DR

This work develops a fast, unified framework for computing steady translating Hele-Shaw bubble configurations in free space, the upper half-plane, and infinite channels under zero surface tension. It fuses boundary-integral equations for conformal mappings with two complex potentials, $w(z)$ and $\tau(z)$, linked by $\tau(z)=w(z)-Uz$, and reduces the free boundary problem to computing mappings from a multiply connected circular domain to canonical slit domains. The method supports very large bubble assemblies, recovers known results, and demonstrates diverse configurations across geometries, including near-wall effects and asymmetric arrays, with controlled numerical accuracy using GMRES-FMM acceleration. This approach enables systematic exploration of complex bubble interactions in constrained geometries and sets the stage for incorporating surface-tension effects in future work.

Abstract

We present a unified numerical method to determine the shapes of multiple Hele-Shaw bubbles in steady motion, and in the absence of surface tension, in three planar domains: free space, the upper half-plane, and an infinite channel. Our approach is based on solving the free boundary problem for the bubble boundaries using a fast and accurate boundary integral method. The main advantage of our method is that it allows for the treatment of a very high number of bubbles. The presented method is validated by recovering some existing results for steady bubbles in channels and free space. Several numerical examples are presented, many of which feature configurations of bubbles that have not appeared in the literature before.

Figures (20)

• Figure 2.1: On the left, a schematic of a Hele-Shaw flow in free space containing an assembly of $m+1=6$ bubbles steadily translating with constant speed $U$ parallel to the $x$-axis. The far field fluid speed is assumed to be $V=1$, and the shapes of the bubble boundaries are to be determined. On the middle and on the right, schematic showing the domains in the $w$-plane and in the $\tau$-plane representing the complex potentials in the fixed and moving frames, respectively.
• Figure 2.2: On the left, a schematic of a Hele-Shaw flow in the upper-half plane containing an assembly of $m=5$ bubbles steadily translating with constant speed $U$ parallel to the $x$-axis. The far field fluid speed is assumed to be $V=1$, and the shapes of the bubble boundaries are to be determined. On the middle and on the right, schematic showing the domains in the $w$-plane and in the $\tau$-plane representing the complex potentials in the fixed and moving frames, respectively.
• Figure 2.3: On the left, a schematic of a Hele-Shaw flow in an infinite channel containing an assembly of $m=5$ bubbles steadily translating with constant speed $U=1.75$ parallel to the $x$-axis. The far field fluid speed is assumed to be $V=1$, and the shapes of the bubble boundaries are to be determined. On the middle and on the right, schematic showing the domains in the $w$-plane and in the $\tau$-plane representing the complex potentials in the fixed and moving frames, respectively.
• Figure 2.4: The circular domain $D_\zeta$ in the auxiliary complex $\zeta$-plane.
• Figure 4.1: The configurations of two bubbles in free space in Example \ref{['ex:spac2']} corresponding to several values of $U$. The area of each bubble is always $\pi$.